4/28/15CMPS 3130/6130 Computational Geometry1 CMPS 3130/6130 Computational Geometry Spring 2015 Shape Matching Carola Wenk A B (B,A)
Are these shapes similar? Translate When are two shapes similar?
Are these shapes similar? Translate Translate & Rotate When are two shapes similar?
Are these shapes similar? Translate Translate & Rotate Translate & Scale When are two shapes similar?
Are these shapes similar? Translate Translate & Rotate Translate & Scale Translate & Reflect When are two shapes similar?
Are these shapes similar? Translate Translate & Rotate Translate & Scale Translate & Reflect Partial Matching When are two shapes similar?
Are these shapes similar? Translate Translate & Rotate Translate & Scale Translate & Reflect Partial Matching When are two shapes similar?
Two geometric shapes, each composed of a number of basic objects such as Given: An application dependant distance measure , e.g., the Hausdorff distance A set of transformations T, e.g., translations, rigid motions, or none Matching Task: Compute min (T(A),B) pointsline segmentstriangles Geometric Shape Matching 4/28/15CMPS 3130/6130 Computational Geometry8
Hausdorff distance Directed Hausdorff distance (A,B) = max min || a-b || Undirected Hausdorff-distance (A,B) = max ( (A,B), (B,A) ) If A, B are discrete point sets, |A|=m, |A|=n In d dimensions: Compute in O(mn) time brute-force In 2 dimensions: Compute in O((m+n) log (m+n) time Compute (A,B) by computing VD(B) and performing nearest neighbor search for every point in A A B (B,A) (A,B) a A b B 4/28/15CMPS 3130/6130 Computational Geometry9
Shape Matching - Applications Character Recognition Fingerprint Identification Molecule Docking, Drug Design Image Interpretation and Segmentation Quality Control of Workpieces Robotics Pose Determination of Satellites Puzzling... 4/28/15CMPS 3130/6130 Computational Geometry10
Hausdorff distance (m 2 n 2 ) lower bound construction for directed Hausdorff distance of line segments under translations 4/28/15CMPS 3130/6130 Computational Geometry11
Comparing Curves and Trajectories 4/28/15CMPS 3130/6130 Computational Geometry12
Polygonal Curves Let f,g:[0,1] R d be two polygonal curves (i.e., piecewise linear curves) What are good distance measures for curves? –Hausdorff distance? –Fréchet distance? f g 4/28/15CMPS 3130/6130 Computational Geometry13
When Are Two Curves „Similar“? Directed Hausdorff distance (A,B) = max min || a-b || Undirected Hausdorff-distance (A,B) = max ( (A,B), (B,A) ) But: Small Hausdorff distance When considered as curves the distance should be large The Fréchet distance takes the continuity of the curves into account A B (B,A) (A,B) a A b B 4/28/15CMPS 3130/6130 Computational Geometry14
F (f,g) = inf max ||f( (t))-g( (t))|| :[0,1] [0,1] t [0,1] where and range over continuous monotone increasing reparameterizations only. Man and dog walk on one curve each They hold each other at a leash They are only allowed to go forward F is the minimal possible leash length f g Fréchet Distance for Curves [F06] M. Fréchet, Sur quelques points de calcul fonctionel, Rendiconti del Circolo Mathematico di Palermo 22: 1-74, /28/15CMPS 3130/6130 Computational Geometry15
Free Space Diagram Let > 0 fixed (eventually solve decision problem) F (f,g) = { (s,t) [0,1] 2 | || f(s) - g(t)|| } white points free space of f and g The free space in one cell is an ellipse. ε g f f 0 1 g >> 4/28/15CMPS 3130/6130 Computational Geometry16
Free Space Diagram Let > 0 fixed (eventually solve decision problem) F (f,g) = { (s,t) [0,1] 2 | || f(s) - g(t)|| } white points free space of f and g The free space in one cell is an ellipse. g f f g 4/28/15CMPS 3130/6130 Computational Geometry17
Free Space Diagram Monotone path encodes reparametrizations of f and g F (f,g) iff there is a monotone path in the free space from (0,0) to (1,1) g f g f 4/28/15CMPS 3130/6130 Computational Geometry18
Compute the Fréchet Distance Solve the optimization problem In practice in O(mn log b) time with binary search and b-bit precision In O(mn log mn) time [AG95] using parametric search (using Cole‘s sorting trick) In O(mn log 2 mn) expected time [CW09] with randomized red/blue intersections [AG95] H. Alt, M. Godau, Computing the Fréchet distance between two polygonal curves, IJCGA 5: 75-91, [CW10] A.F. Cook IV, C. Wenk, Geodesic Fréchet Distance Inside a Simple Polygon, ACM TALG 7(1), 19 pages, Solve the decision problem F (f,g) in O(mn) time: Find monotone path using DP: On each cell boundary compute the interval of all points that are reachable by a monotone path from (0,0) Compute a monotone path by backtracking f g /28/15CMPS 3130/6130 Computational Geometry
Navigation systems answer shortest path queries based on travel times on road segments Usually based on static travel-time weights derived from speed limits Goal: Develop a navigation system which answers routing queries based on the current traffic situation A B Current traffic situation: Database of current travel times per road segment Use GPS trajectory data from vehicle fleets GPS Trajectories for Dynamic Routing 4/28/15CMPS 3130/6130 Computational Geometry20
GPS Trajectories from Vehicles Road map of Athens GPS trajectoriy Corresponding path in the road map 1)Measurement error: GPS points generally do not lie on the road map 2)Sampling error: The GPS trajectory is a by-product and is usually sampled every 30s The GPS trajectory does not lie on the road map Map matching: Find a path in the graph which corresponds to the GPS trajectory (curve). Find a path in the graph with minimal distance to the GPS curve (partial matching) 214/28/15CMPS 3130/6130 Computational Geometry
Free Space Surface Glue the free space diagrams FD i,j together according to adjacency information in G Free space surface of f and G G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , /28/15CMPS 3130/6130 Computational Geometry22
Free Space Surface Task: Find a monotone path in the free space surface that –starts in a lower left corner –and ends in an upper right corner G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , /28/15CMPS 3130/6130 Computational Geometry23
Sweep G Sweep all FD i,j with a sweep line from left to right [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , /28/15CMPS 3130/6130 Computational Geometry24
G Sweep [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , Sweep all FD i,j with a sweep line from left to right 4/28/15CMPS 3130/6130 Computational Geometry25
G Sweep [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , Sweep all FD i,j with a sweep line from left to right 4/28/15CMPS 3130/6130 Computational Geometry26
G Sweep [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , Sweep all FD i,j with a sweep line from left to right 4/28/15CMPS 3130/6130 Computational Geometry27
G Sweep [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , Sweep all FD i,j with a sweep line from left to right 4/28/15CMPS 3130/6130 Computational Geometry28
G Sweep [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , Sweep all FD i,j with a sweep line from left to right 4/28/15CMPS 3130/6130 Computational Geometry29
G Sweep [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , Sweep all FD i,j with a sweep line from left to right 4/28/15CMPS 3130/6130 Computational Geometry30
Compute Reachable Points G During the sweep: Compute points on the free space surface, to the left of the sweep line, which are reachable by a monotone path from a lower left corner. [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , /28/15CMPS 3130/6130 Computational Geometry31
G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , Compute Reachable Points During the sweep: Compute points on the free space surface, to the left of the sweep line, which are reachable by a monotone path from a lower left corner. 4/28/15CMPS 3130/6130 Computational Geometry32
G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , Compute Reachable Points During the sweep: Compute points on the free space surface, to the left of the sweep line, which are reachable by a monotone path from a lower left corner. 4/28/15CMPS 3130/6130 Computational Geometry33
G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , Compute Reachable Points During the sweep: Compute points on the free space surface, to the left of the sweep line, which are reachable by a monotone path from a lower left corner. 4/28/15CMPS 3130/6130 Computational Geometry34
Update Reachable Points During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , G 4/28/15CMPS 3130/6130 Computational Geometry35
[AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , G During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry36
[AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , G During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry37
[AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , G During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry38
[AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , G During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry39
G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry40
G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry41
G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry42
G [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , During the sweep: –Update reachable points Dijkstra-style –Use a data structure which supports reachability queries in the free space surface Update Reachable Points 4/28/15CMPS 3130/6130 Computational Geometry43
G Backtracking After the sweep: –Construct a monotone path via backtracking [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , /28/15CMPS 3130/6130 Computational Geometry44
Map-Matching Algorithm for decision problem takes O(mn log(mn)) time and O(mn) space. Optimization problem with parametric search: O(mn log 2 (mn)) time [AERW03] H. Alt, A. Efrat, G. Rote, C. Wenk, Matching Planar Maps, J. of Algorithms 49: , [BPSW05] S. Brakatsoulas, D. Pfoser, R. Salas, C. Wenk, On Map-Matching Vehicle Tracking Data, VLDB , [WSP06] C. Wenk, R. Salas, D. Pfoser, Adressing the Need for Map-Matching Speed…, SSDBM: , /28/15CMPS 3130/6130 Computational Geometry45